Question

2. Use the principle of mathematical induction to prove the following statements (a) \( \sum_{i=1}^{n}(2 i-1)^{2}=\frac{n(2 n-1)(2 n=1)}{3} \). (b) \( 2^{2 n+1}+1 \) is divisible by 3.6 (c) If \( A_{1}, A_{2}, \ldots, A_{n} \) are subsets of the universal set \( U \), then \( \left(\bigcap_{i=1}^{n} A_{i}\right)^{\prime}=\bigcup_{i=1}^{n} A_{i}^{\prime} \), where \( A_{i}^{\prime} \) denotes the complement of the set \( A_{i} \).

          2. Use the principle of mathematical induction to prove the following statements
(a) \( \sum_{i=1}^{n}(2 i-1)^{2}=\frac{n(2 n-1)(2 n=1)}{3} \).
(b) \( 2^{2 n+1}+1 \) is divisible by 3.6
(c) If \( A_{1}, A_{2}, \ldots, A_{n} \) are subsets of the universal set \( U \), then \( \left(\bigcap_{i=1}^{n} A_{i}\right)^{\prime}=\bigcup_{i=1}^{n} A_{i}^{\prime} \), where \( A_{i}^{\prime} \) denotes the complement of the set \( A_{i} \).

2. Use the principle of mathematical induction to prove the following statements
(a) ∑i=1^n(2 i-1)^2=(n(2 n-1)(2 n=1))/(3).
(b) 2^2 n+1+1 is divisible by 3.6
(c) If A1, A2, …, An are subsets of the universal set U, then (⋂i=1^n Ai)^'=⋃i=1^n Ai^', where Ai^' denotes the complement of the set Ai.

Added by Selasi J.

Precalculus Functions and Graphs

Mark Dugopolski 4th Edition

Chapter 11

Instant Answer

Solved by Expert Eric Carlsen

Step 1

Base Case: For n=1, the left side is (2*1-1)^2 = 1 and the right side is 1*(2*1-1)*(2*1+1)/3 = 1. So, the base case holds. Inductive Step: Assume the statement is true for n=k, i.e., \( \sum_{i=1}^{k}(2 i-1)^{2}=\frac{k(2 k-1)(2 k+1)}{3} \). We need to prove it Show more…

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